A review and re-interpretation of a group-sequential approach to sample size re-estimation in two-stage trials.

Bowden J, Mander A - Pharm Stat (2014)

Bottom Line:
In this paper, we review the adaptive design methodology of Li et al. (Biostatistics 3:277-287) for two-stage trials with mid-trial sample size adjustment.We argue that it is closer in principle to a group sequential design, in spite of its obvious adaptive element.Several extensions are proposed that aim to make it even more attractive and transparent alternative to a standard (fixed sample size) trial for funding bodies to consider.

fig06: Left: expected sample size of the LSW (design 1) and modified LSW (design 2) using theoretical calculation (black) and using empirical simulation (red). Right: type I error rate inflation (above the nominal 0.025 level) when the data is used to estimate σ under the modified LSW design. Design 4 highlighted in red.

Mentions:
We assume that σ is known in the calculations used to both find our designs and report their operating characteristics. The simple mathematical formulae would not work if σ were treated as a random variable. However, in practice one will need to estimate it from the data to implement any of the design proposals. It is important therefore to verify that this estimation does not cause a design's true operating characteristics to differ substantially from its theoretical counterpart. Figure 6 (left) shows the expected sample size of Design's 1 and 2 as a function of δ using (a) theoretical calculation (i.e. using formulae from Table S1) and (b) via simulation (incorporating estimation of σ separately at stage 1 and 2). To clarify, treatment and control group data were simulated from equation (1) for specific values of μx, μy and n1, but with a common value of σ = 20. This defined the theoretical value of δ. A pooled estimate for σ, was then obtained from these two populations and was estimated as . If the trial proceeded to stage 2, σ was re-estimated from the n2(z1) additionally simulated patients in each arm in the same manner, and used to calculate , z2 and z for equation (2). The difference between the theoretical expected sample size and those obtained in practice (with estimation of σ) is tiny, which is re-assuring. The theoretical and practical power curves for these designs are also near identical (results not shown). However, it is of crucial importance to check that the type I error rate is not drastically inflated (i.e. the power when δ=0).

fig06: Left: expected sample size of the LSW (design 1) and modified LSW (design 2) using theoretical calculation (black) and using empirical simulation (red). Right: type I error rate inflation (above the nominal 0.025 level) when the data is used to estimate σ under the modified LSW design. Design 4 highlighted in red.

Mentions:
We assume that σ is known in the calculations used to both find our designs and report their operating characteristics. The simple mathematical formulae would not work if σ were treated as a random variable. However, in practice one will need to estimate it from the data to implement any of the design proposals. It is important therefore to verify that this estimation does not cause a design's true operating characteristics to differ substantially from its theoretical counterpart. Figure 6 (left) shows the expected sample size of Design's 1 and 2 as a function of δ using (a) theoretical calculation (i.e. using formulae from Table S1) and (b) via simulation (incorporating estimation of σ separately at stage 1 and 2). To clarify, treatment and control group data were simulated from equation (1) for specific values of μx, μy and n1, but with a common value of σ = 20. This defined the theoretical value of δ. A pooled estimate for σ, was then obtained from these two populations and was estimated as . If the trial proceeded to stage 2, σ was re-estimated from the n2(z1) additionally simulated patients in each arm in the same manner, and used to calculate , z2 and z for equation (2). The difference between the theoretical expected sample size and those obtained in practice (with estimation of σ) is tiny, which is re-assuring. The theoretical and practical power curves for these designs are also near identical (results not shown). However, it is of crucial importance to check that the type I error rate is not drastically inflated (i.e. the power when δ=0).

Bottom Line:
In this paper, we review the adaptive design methodology of Li et al. (Biostatistics 3:277-287) for two-stage trials with mid-trial sample size adjustment.We argue that it is closer in principle to a group sequential design, in spite of its obvious adaptive element.Several extensions are proposed that aim to make it even more attractive and transparent alternative to a standard (fixed sample size) trial for funding bodies to consider.